Euclidean operator radius inequalities of a pair of bounded linear operators and their applications

Abstract: We obtain several sharp lower and upper bounds for the Euclidean operator radius of a pair of bounded linear operators defined on a complex Hilbert space. As applications of these bounds we deduce a chain of new bounds for the classical numerical radius of a bounded linear operator which improve on the existing ones. In particular, we prove that for a bounded linear operator $A,$ [\frac{1}{4} |A^A+AA^|+\frac{\mu}{2}\max {|\Re(A)|,|\Im(A)|} \leq w^2(A) \, \leq \, w^2( |\Re(A)| +i |\Im(A)|),] where $\mu= \big| |\Re(A)+\Im(A)|-|\Re(A)-\Im(A)|\big|.$ This improve the existing upper and lower bounds of the numerical radius, namely, [ \frac14 |A^A+AA^|\leq w^2(A) \leq \frac12 |A^A+AA^|. ]